
Compared to the existing decorrelated score approach [1]}, our methodological innovation is to develop an explicit bias correction step to remove the approximation bias in (REF ) induced by the smoothed surrogate risk. From the theoretical aspect, our test statistic \(\widehat{U}_n\)  is rescaled by \((n\delta )^{1/2}\)  rather than the classical \(n^{1/2}\)  factor, which leads to the non-standard rate of the decorrelated score not only under the null but also under local alternatives; see Section .
